# Fractional Calculus and its Connection to the Tautochrone

## Abstract

In 1695, the mathematician Guillaume de L'Hospital wrote a letter to the Gottfried Leibniz asking what it would mean to take a fractional order derivative. Leibniz responded by saying it would be a problem for future generations. For centuries, very little progress was made, leaving Fractional Calculus a relatively untapped field. With most major contributions occurring in the last one-hundred years. In this paper, we will examine the fundamental aspects of Fractional Calculus, and demonstrate how the modern definitions of the Fractional Integral naturally arise from solving the classic Tautochrone problem: finding a curve such that the time it takes an object to fall along this path is independent of its initial position. We then consider a generalization of the Tautochrone by investigating the case when time is dependent on initial position. The result is a theorem building on the work published by Munoz and Fernandez-Anaya. We will also examine the Mittag-Leffler function, and how it arises in the solution to Abel’s Integral Equation of the second kind.

Fractional Calculus and its Connection to the Tautochrone

In 1695, the mathematician Guillaume de L'Hospital wrote a letter to the Gottfried Leibniz asking what it would mean to take a fractional order derivative. Leibniz responded by saying it would be a problem for future generations. For centuries, very little progress was made, leaving Fractional Calculus a relatively untapped field. With most major contributions occurring in the last one-hundred years. In this paper, we will examine the fundamental aspects of Fractional Calculus, and demonstrate how the modern definitions of the Fractional Integral naturally arise from solving the classic Tautochrone problem: finding a curve such that the time it takes an object to fall along this path is independent of its initial position. We then consider a generalization of the Tautochrone by investigating the case when time is dependent on initial position. The result is a theorem building on the work published by Munoz and Fernandez-Anaya. We will also examine the Mittag-Leffler function, and how it arises in the solution to Abel’s Integral Equation of the second kind.